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CTET Paper II (Mathematics and Science) Sep 2016 Solved Paper

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Question : 46 of 150

Marks: +1, -0

One of the factors of x^{4} + 4 is

$x^{2} - 2x + 2$
$x^{2} - 2$
$x^{2} + 2$
$x^{2} + 2x - 2$

Solution:

Concept:

To factorize

x^{4} + 4

, we rewrite it as a difference of squares by adding and subtracting

4x^{2}

.

Explanation:

Start with

x^{4} + 4

. Add and subtract

4x^{2}

to get:

x^{4} + 4 = x^{4} + 4 + 4x^{2} - 4x^{2}

.

Group as

(x^{4} + 4 + 4x^{2}) - 4x^{2}

. This becomes

(x^{2} + 2)^{2} - (2x)^{2}

.

Now apply

a^{2} - b^{2} = (a + b)(a - b)

:

(x^{2} + 2 - 2x)(x^{2} + 2 + 2x)

.

So the factors are

x^{2} - 2x + 2

and

x^{2} + 2x + 2

.

Answer:

Option A:

x^{2} - 2x + 2

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